Let \(R\ge R_0\) .
Since \(\infty \)  is not an asymptotic value, Lemma 2.1 in [1]} asserts that \(f^{-1} ({\mathbb {D}}_R^* )\)  is a countable union of simply connected bounded components
\(U_j(R)\) , \(j \ge 1\) , each of which contains exactly one pole \(a_j\in U_j(R)\) . When \(R\ge R_0\)  is fixed then we often denote these components simply \(U_j\) .
